Why you can't make Kevlar without modern equipment.
The manufacture of Kevlar has a few roadblocks. Kevlar is made from p-Phenylenediamine and Terephthaloyl Chloride.
The p-Phenylenediamine part would be hard but doable. You need Aluminum Chloride as part of the production chain. The production of Aluminum Chloride typically requires electrolysis to isolate the aluminum using the most cost efficient method... that said, there are expensive, not electrolysis methods that you could use to get around this. All the other materials and processes used to make p-Phenylenediamine were available by the late 1700s.
The Terephthaloyl Chloride is the much harder part because you need to run naphtha through the catalytic reforming process to get to p-Xylene. This step requires complex and expensive refining equipment, and there is no simple substitute for this step that I can find. p-Xylene is needed at multiple steps in the production of Terephthaloyl Chloride.
This said, Kevlar might make good body armor, against bullets, but it does not have the best puncture resistance (which is what you actually need on a pre-modern battlefield). While it might outperform a linen based gambison, it will not protect you better that steel mail or plate armor would against most threats you will face. It is also not a good materials to use in bow construction because it is too stretchy and does not have a lot of snap back like certain other materials.
The modern synthetic materials you really want to replace are resin-fiberglass composites and nylon.
Natural Resin-fiberglass composites
The main reason modern bows shoot faster is because they are reinforced with fiberglass. Fiberglass springs back from deformation much faster than and wood or metal allowing a fiberglass backed bow arm to shoot nearly twice as fast as traditional bow of similar design.
This said, man-made resins and fiber glasses have naturally occurring counterparts. Asbestos is a naturally occurring fiberglass that people have been using for over 6000 years. In fact, it has a higher tensile strength than most man-made fiber glasses; so, as long as you don't mind occasionally needed to replace some dead bowyers due to its toxicity, it is a great material to use for this.
As for resins, there are many naturally occurring options made from various tree saps to pick from. Though you will specifically want to be looking at one of many possible "copal resin" trees. These trees have saps that when boiled, produce a hard amber like plastic very similar to the resins used in fiberglass composites today. While copal resins will not harden as quickly as modern epoxy resins, they can be very tough once set, and have also been in use for thousands of years.
That said, resin-fiberglass composites are good for more than just bows. Resin-fiberglass can also be used to improve the construction of ships, wagons, body armor, and tool handles just to name a few.
Natural Composite to Substitute for Nylon
As for the bowstring itself, the most important quality of a good bow string is that it does not stretch under tension. Modern bow strings are typically made from Nylon (Not Kevlar) specifically because Nylon can take significant load without stretching compared to other synthetic fibers.
Nylon is much easier to make than Kevlar too, and there are so many different variants of it, that there are a few that can be made without any complex equipment. That said, the ancient world already had a super material that may have been at least as good.
Like Nylon, cords made from horse or human hair are highly resistant to streching under load. Some of history's best bow cording material was invented by the Romans using a combination of animal sinew and human or horse hair reverse twisted together and reinforced with hide glue. Not only was this composite cording very comparable to the nylon bow strings we use today, but it does not require any complex chemistry to make.
